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Full text of "Men Of Mathematics"

THE DOUBTER
and it fills me with a bitter grief, to see a man, whose glory is
without flaw, let himself be driven by the well justified feeling
of his own worth to utterances whose injurious effect upon
others he seems not to perceive.
'But enough of these things, on which I have touched only to
explain to you the reason why I can no longer take the same joy
that I used to take in my teaching, even if my health were to
permit me to continue it a few years longer. But you must not
speak of it; I should not like others, who do not know me as wefl
as you, to see in what I say the expression of a sentiment which
is in fact foreign to^me.5
Weierstrass was seventy and in poor health when he wrote
this. Could he have lived till to-day he would have seen his own
great system still flourishing like the proverbial green bay tree.
Kronecker's doubts have done much to instigate a critical re-
examination of the foundations of all mathematics, but they
have not yet destroyed analysis. They go deeper, and if any-
thing of far-reaching significance is to be replaced by something
firmer but as yet unknown, it seems likely that a good part of
Kronecker*s own work will go too, for the critical attack which
he foresaw has uncovered weaknesses where he suspected
nothing. Time makes fools of us all. Our only comfort is that
greater shall come after us.
Kronecker's 'revolution', as his contemporaries called his sub-
versive assault on analysis, would banish all but the positive
integers from mathematics. Geometry since Descartes has been
largely an affair of analysis applied to ordered pairs, triples, ..
of real numbers (the 'numbers' which correspond to the dist-
ances measured on a given straight line from a fixed point on the
line); hence it too would come under the sway of Kronecker's
programme. So familiar a concept as that of a negative integer,
 2 for instance, would not appear in the mathematics Kron-
ecker prophesied, nor would common fractions.
Irrationals, as Weierstrass points out, roused Kroneckefs
special displeasure. To speak of xz  2 = 0 having a root would
be meaningless. AH of these dislikes and objections are of course
themselves meaningless unless they can be backed by a definite
programme to replace what is rejected,
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